From 37ac3decf6fca2cec79cfe205e52c5fe13d17fd0 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 28 Jun 2019 14:52:29 -0400
Subject: fixed Tc in the figures and fixed a minor mistake in the elastic
 susceptibility caluclation

---
 main.tex | 12 ++++++------
 1 file changed, 6 insertions(+), 6 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index c92400b..4264884 100644
--- a/main.tex
+++ b/main.tex
@@ -181,7 +181,7 @@ which implicitly gives $\eta$ as a functional of $\epsilon_\X$. Though this cann
     \bigg(\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}\bigg)^{-1}
     &=\frac{\delta\eta_j^{-1}[\eta](x)}{\delta\eta_i(x')}
     =-\frac2b\frac{\delta^2F_\o}{\delta\eta_i(x)\delta\eta_j(x')} \\
-    &=-\frac2b\chi^{-1}(x,x')-\frac{b}{2\lambda_\X}\delta(x-x')
+    &=-\frac2b\chi_{ij}^{-1}(x,x')-\frac{b}{2\lambda_\X}\delta_{ij}\delta(x-x')
   \end{aligned}
   \label{eq:inv.func}
 \end{equation}
@@ -191,25 +191,25 @@ It follows from \eqref{eq:implicit.eta} and \eqref{eq:inv.func} that the suscept
   \begin{aligned}
     \chi_{\X ij}^{-1}(x,x')
     &=\frac{\delta^2F}{\delta\epsilon_{\X i}(x)\delta\epsilon_{\X j}(x')} \\
-    &=\lambda_\X\delta(x-x')+
+    &=\lambda_\X\delta_{ij}\delta(x-x')+
     b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}
     +\frac12b\int dx''\,\epsilon_{\X k}(x'')\frac{\delta^2\eta_k(x)}{\delta\epsilon_{\X i}(x')\delta\epsilon_{\X j}(x'')} \\
     &\qquad+\int dx''\,dx'''\,\frac{\delta^2F_\o}{\delta\eta_k(x'')\delta\eta_\ell(x''')}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)}\frac{\delta\eta_\ell(x''')}{\delta\epsilon_{\X j}(x')}
     +\int dx''\,\frac{\delta F_\o}{\delta\eta_k(x'')}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)\delta\epsilon_{\X j}(x')} \\ 
-    &=\lambda_\X\delta(x-x')+
+    &=\lambda_\X\delta_{ij}\delta(x-x')+
     b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}
     -\frac12b\int dx''\,dx'''\,\bigg(\frac{\partial\eta_k(x'')}{\partial\epsilon_{\X\ell}(x''')}\bigg)^{-1}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)}\frac{\delta\eta_\ell(x''')}{\delta\epsilon_{\X j}(x')} \\ 
-    &=\lambda_\X\delta(x-x')+
+    &=\lambda_\X\delta_{ij}\delta(x-x')+
     b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}
     -\frac12b\int dx''\,\delta_{i\ell}\delta(x-x'')\frac{\delta\eta_\ell(x'')}{\delta\epsilon_{\X j}(x')} 
-    =\lambda_\X\delta(x-x')+
+    =\lambda_\X\delta_{ij}\delta(x-x')+
     \frac12b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')},
   \end{aligned}
 \end{equation}
 \end{widetext}
 whose Fourier transform follows from \eqref{eq:inv.func} as
 \begin{equation}
-  \chi_{\X ij}(q)=\frac1{\lambda_\X}+\frac{b^2}{4\lambda_\X^2}\chi_{ij}(q).
+  \chi_{\X ij}(q)=\frac{\delta_{ij}}{\lambda_\X}+\frac{b^2}{4\lambda_\X^2}\chi_{ij}(q).
   \label{eq:elastic.susceptibility}
 \end{equation}
 At $q=0$, which is where the stiffness measurements used here were taken, this predicts a cusp in the elastic susceptibility of the form $|\tilde r-\tilde r_c|^\gamma$ for $\gamma=1$.
-- 
cgit v1.2.3-70-g09d2

